Using reverse arrangement for trend test in statistical process control for manufacture of semiconductor integrated circuits

ABSTRACT

A method for manufacturing semiconductor devices or other types of devices and/or entities. The method includes providing a process (e.g., etching, deposition, implantation) associated with a manufacture of a semiconductor device/The method includes collecting a plurality information (e.g., data) having a non-monotonic trend of at least one parameter associated with the process over a determined period. The method includes processing the plurality of information having the non-monotonic trend. The method includes detecting an increasing or a decreasing trend from the processed plurality of information having the non-monotonic trend. The method includes performing an action based upon at least the detected increasing or decreasing trend.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application claims priority to Chinese Application No. 200610025382.9; filed on Mar. 28, 2006; commonly assigned, and of which is hereby incorporated by reference for all purposes.

COPYRIGHT NOTICE

Certain portions of the present specification include computer codes, where notice is hereby given. All rights have been reserved under Copyright for such computer codes, by ©2004 and 2005 Semiconductor Manufacturing International (Shanghai) Corporation, which is the present assignee.

BACKGROUND OF THE INVENTION

The present invention is directed to integrated circuits and their processing for the manufacture of semiconductor devices. In particular, the invention provides a method and system for monitoring and controlling process related information for the manufacture of semiconductor integrated circuit devices. More particularly, the invention provides a method and system using a reverse arrangement process for a trend test(s) for statistical process control used in the manufacture of semiconductor integrated circuit devices. But it would be recognized that the invention has a much broader range of applicability.

Integrated circuits have evolved from a handful of interconnected devices fabricated on a single chip of silicon to millions of devices. Conventional integrated circuits provide performance and complexity far beyond what was originally imagined. In order to achieve improvements in complexity and circuit density (i.e., the number of devices capable of being packed onto a given chip area), the size of the smallest device feature, also known as the device “geometry”, has become smaller with each generation of integrated circuits.

Increasing circuit density has not only improved the complexity and performance of integrated circuits but has also provided lower cost parts to the consumer. An integrated circuit or chip fabrication facility can cost hundreds of millions, or even billions, of U.S. dollars. Each fabrication facility will have a certain throughput of wafers, and each wafer will have a certain number of integrated circuits on it. Therefore, by making the individual devices of an integrated circuit smaller, more devices may be fabricated on each wafer, thus increasing the output of the fabrication facility. Making devices smaller is very challenging, as each process used in integrated fabrication has a limit. That is to say, a given process typically only works down to a certain feature size, and then either the process or the device layout needs to be changed. Additionally, as devices require faster and faster designs, process limitations exist with certain conventional processes, including monitoring techniques, materials, and even testing techniques.

An example of such processes include ways of monitoring process related functions during the manufacture of integrated circuits, commonly called semiconductor devices. Such monitoring process is often desired for continuously improving quality and productivity to stay competitive. As merely an example, statistical process control (SPC) has been playing an important role in conventional industries. It is a procedure in which data are collected, organized, analyzed and interpreted. Actions are requested to identify root causes and to implement solutions so a process can be maintained at its desired level or be improved to a higher level. SPC makes use of statistical signals to identify sources of variation, to correct identified variation causes therefore to improve performance, and to maintain control of processes. Variations are classified as common (random or chance) and special (or assignable) causes in general [1]. Common causes denote the many sources of variation within a process that is in statistical control. Special causes refer to any factors causing variation that cannot be adequately explained by a single distribution. A process in statistical control operates with less variability than a process having special causes. Unless all the special causes of variance are identified and corrected, they will continue to affect the process outputs in unpredictable and undesirable ways.

Control charts (which are trend charts with control limits) are often used to monitor selected parameters, which have important quality characteristics. Various run tests have been developed to identify if there is any pattern in the data points. Western Electric developed five run tests [2]; they are 1) 1 point beyond 3 sigma, 2) 2 out 3 successive points beyond 2 sigma, 3) 4 out of 5 successive points beyond 1 sigma, 4) 15 successive points not within 1 sigma of center line, and 5) 8 successive points on the same side and not within 1 sigma of center line. Later in about 1986, Nelson developed additional 3 rules [3]: 1) 9 successive points on same side of center line, 2) 6 successive points steadily increasing or decreasing, and 3) 14 successive points alternating up and down.

The run test of 6 consecutive points increasing or decreasing, proposed by Nelson is a special test of the trend pattern, indicating an instable process. It is usually assumed that the change will be monotonic and is either increasing or decreasing over time. The ease of such monotonic trend test becomes popular due to the practical values. However, this test obviously cannot detect all possible trends and we should be aware that the change may be non-monotonic (i.e., fluctuating). Other limitations also exist with these conventional techniques. These and other limitations are described throughout the present specification and more particularly below.

From the above, it is seen that an improved technique for manufacturing semiconductor devices is desired.

BRIEF SUMMARY OF THE INVENTION

According to the present invention, techniques directed to integrated circuits and their processing for the manufacture of semiconductor devices are provided. In particular, the invention provides a method and system for monitoring and controlling process related information for the manufacture of semiconductor integrated circuit devices. More particularly, the invention provides a method and system using a reverse arrangement process for a trend test(s) for statistical process control used in the manufacture of semiconductor integrated circuit devices. But it would be recognized that the invention has a much broader range of applicability.

In further background, we identified that other forms of tends such as non-monotonic trends from, for example, parts wearing-out, and other physical conditions, and the like. The present method and system uses a powerful and special test called “Reverse Arrangement Test” (RAT) to identify monotonic as well as non-monotonic increasing or decreasing trends for any possible number (>=6) of points under test according to a specific embodiment. As merely example, we have also provided cases that reported using the RAT test to show its contributions.

In a specific embodiment, the present invention provides a method for manufacturing semiconductor devices or other types of devices and/or entities. The method includes providing a process (e.g., etching, deposition, implantation) associated with a manufacture of a semiconductor device/The method includes collecting a plurality information (e.g., data) having a non-monotonic trend of at least one parameter associated with the process over a determined period. The method includes processing the plurality of information having the non-monotonic trend. The method includes detecting an increasing or a decreasing trend from the processed plurality of information having the non-monotonic trend. The method includes performing an action based upon at least the detected increasing or decreasing trend.

In an alternative specific embodiment, the present invention provides a system for manufacturing semiconductor devices. In a preferred embodiment, the system has one or more memories, e.g., hard disk drives, random access memory, Flash memories, static memories. Various computer codes are provided to carry out functionality described herein. The system has one or more codes directed to initiating a process associated with a manufacture of a semiconductor device. The system also has one or more codes directed to collecting a plurality information having a non-monotonic trend of at least one parameter associated with the process over a determined period. The system has one or more codes directed to processing the plurality of information having the non-monotonic trend. One or more codes is also directed to detecting an increasing or a decreasing trend from the processed plurality of information having the non-monotonic trend. One or more codes is directed to outputting a code to perform an action based upon at least the detected increasing or decreasing trend.

Additionally, one or more limitations of conventional trend test in SPC practice has also been identified. The six consecutive increasing or decreasing points cannot detect non-monotonic increasing or decreasing trend, which is frequently encountered in practice, as we have identified. In a specific embodiment, the present method and system provides a RAT (reverse arrangement test) test to replace, at least in part and/or supplement, conventional SPC trend test rule in order to detect non-monotonic increasing or decreasing trends. A theory of RAT is reviewed and we point out the errors in the tables from a well-known and most frequently referenced paper on RAT by Mann [5]. The corrected tables of accumulated probability for each total reverse arrangement for n=3 to 12 are presented. For the first time in literature, we illustrate a flaw of RAT for observations with identical values and propose to check tied data before applying RAT. Examples show that 7 non-monotonic increasing points can only be detected by RAT, while none of the current WECO rules can detect such abnormal pattern. Applications of RAT in IC SPC and WLRC (Wafer Level Reliability Control) show RAT is more sensitive than conventional monotonic trend tests. Further details of the present invention can be found throughout the present specification and more particularly below.

In a specific embodiment, the invention can also include one or more of the following features.

1. A method (RAT) to replace the monotonic trend test in conventional SPC practice according to a specific embodiment. The RAT not only detects monotonic trend but also for non-monotonic trend.

2. A heuristic is presented by a flow chart for the proposed RAT procedure for its application in semiconductor SPC according to an alternative embodiment of the present invention. We also depict the disposition on tied data.

3. The method and system also introduced a table on critical R for up or down trend test for practical use according to yet an alternative embodiment of the present invention. However, different false alarm rate criteria could be used and the table can be changed accordingly.

4. In an alternative embodiment, the present method and system can be used to correct the error on accumulated probability for a certain R in Table 1 of the original paper by Mann [5].

As will be appreciated, the present method and system and related description herein are purely illustrative and are not to be limited with RAT. Many if not all tests of randomness available in statistical literature can be used to serve to detect non-monotonic trends too, such as Cox-Stuart test (Cox and Stuart, Some Quick tests for trend in location and dispersion. Biometrika, 42, 80-95, 1955), and Daniels Test (Daniels, H. E. Rank Correlation and Population Models, Journal of the Royal Statistical Society (B), 12, 171-181, 1950). Of course, there can be other variations, modifications, and alternatives.

Many benefits are achieved by way of the present invention over conventional techniques. For example, the present technique provides an easy to use process that relies upon conventional technology. In some embodiments, the method provides higher device reliability and performance. Depending upon the embodiment, one or more of these benefits may be achieved. These and other benefits will be described in more throughout the present specification and more particularly below.

Various additional objects, features and advantages of the present invention can be more fully appreciated with reference to the detailed description and accompanying drawings that follow.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a simplified flow chart illustrating a method according to an embodiment of the present invention;

FIG. 2 is a table (Table 1) illustrating a frequency distribution of sigma for values of N from 1 to 6;

FIG. 3 is a table (Table 2) illustrating an accumulated frequency and corresponding probability for each R and signal;

FIG. 4 is a table (Table 4) listing minimum and maximum total reverse arrangements for up and down trend test and their corresponding p-values (i.e., the false alarm rate);

FIG. 5 is a simplified control chart for example 3 according to an embodiment of the present invention;

FIG. 6 is a simplified control chart for example 4 according to an embodiment of the present invention;

FIG. 7 is a simplified chart applying RAT according to an embodiment of the present invention;

FIG. 8 is a simplified flow chart illustrating a method of RAT according to an embodiment of the present invention;

FIG. 9 is a simplified computer system according to an embodiment of the present invention; and

FIG. 10 is a simplified block diagram of a computer system according to an embodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

According to the present invention, techniques directed to integrated circuits and their processing for the manufacture of semiconductor devices are provided. In particular, the invention provides a method and system for monitoring and controlling process related information for the manufacture of semiconductor integrated circuit devices. More particularly, the invention provides a method and system using a reverse arrangement process for a trend test(s) for statistical process control used in the manufacture of semiconductor integrated circuit devices. But it would be recognized that the invention has a much broader range of applicability. Details of the present invention can be found throughout the present specification and more particularly below.

In a specific embodiment, the present invention provides a method for manufacturing semiconductor devices or other types of devices and/or entities, which has been identified below (See, FIG. 1).

1. Provide a process (e.g., etching, deposition, implantation) (Step 101) associated with a manufacture of a semiconductor device;

2. Collect (Step 103) a plurality of information (e.g., data) having a non-monotonic trend of at least one parameter associated with the process over a determined period;

3. Store (Step 105) the plurality of information in memory;

4. Process (Step 107) the plurality of information having the non-monotonic trend;

5. Detect (Step 109) an increasing or a decreasing trend from the processed plurality of information having the non-monotonic trend;

6. Perform (Step 111) an action based upon at least the detected increasing or decreasing trend; and

7. Perform (Step 113) other steps, as desired.

The above sequence of steps provides methods according to an embodiment of the present invention. As shown, the method uses a combination of steps including a way of performing a SPC process according to an embodiment of the present invention. Many other methods and system are also included. Of course, other alternatives can also be provided where steps are added, one or more steps are removed or repeated, or one or more steps are provided in a different sequence without departing from the scope of the claims herein. Additionally, the various methods can be implemented using a computer code or codes in software, firmware, hardware, or any combination of these. Depending upon the embodiment, there can be other variations, modifications, and alternatives. Before discussing specific aspects of the present invention, we have described in details of various conventional techniques that we have evaluated.

We understand that together with his famous correlation coefficient τ in Eq. (1), M. G. Kendall firstly introduced the concept of reverse arrangement in 1938 [4]. Kendall's τ was defined as:

$\begin{matrix} {\tau = \frac{{actual}\mspace{14mu} {score}\mspace{14mu}\sum}{{maximum}\mspace{14mu} {possible}\mspace{14mu} {score}}} & (1) \end{matrix}$

Kendall explained the “score” in Eq. (1) by a numerical example owith10 arbitrary ranking numbers:

-   -   4 7 2 10 3 6 8 1 5 9

Score=+1 if the second number of a pair is greater than the first one. Score=−1 is then defined oppositely. Consider the first number, i.e., 4. There are 9 pairs for the remaining nine numbers associated with 4 and, by the definition on score, we have: (4, 7)→+1, (4, 2)→−1, (4, 10)→+1, (4, 3)→4-1, (4, 6)→+1, (4, 8)→+1, (4, 1)→−1, (4, 5)−+1, and (4, 9)−+1. The sum of these nine scores is then: Σ(+1−1+1−1+1+1−1+1+1)=+3.

Consider the second number, i.e., 7. There are 8 pairs and the scores are (−1, +1, −1, −1, +1, −1, −1, +1). The sum is −2. Continue doing such scoring for the first nine numbers and the nine scores are (+3, −2, +5, −6, +3, 0, −1, +2, +1). The sum of these score is +5, which is the Σ in the numerator of Eq. (1). If the 10 numbers are in the ascending order (1, 2, 3, . . . , 10), we obtain the maximum score 45, which is the denominator of Eq. (1). Therefore, the correlation coefficient τ is

$\begin{matrix} {\tau = {\frac{{actual}\mspace{14mu} {score}\mspace{14mu}\sum}{{maximum}\mspace{14mu} {possible}\mspace{14mu} {score}} = {\frac{5}{45} = {+ 0.11}}}} & (2) \end{matrix}$

The maximum possible score for n individuals is

${\left( {n - 1} \right) + \left( {n - 2} \right) + \ldots + 1} = {\frac{n\left( {n - 1} \right)}{2}.}$

Hence, we have

$\begin{matrix} {\tau = \frac{\Sigma}{{n\left( {n - 1} \right)}/2}} & (3) \end{matrix}$

In the same paper [4], Kendall introduced a convenient method for calculations by means of the reverse arrangement named later by Henry B. Mann [5]. From the set of observations x₁, x₂, . . . , x_(N), the reverse arrangement is defined as

$\begin{matrix} {h_{ij} = \left\{ {\begin{matrix} 1 & {{if}\mspace{14mu} x_{i}{\langle x_{j}}} \\ 0 & {otherwise} \end{matrix}{Then}} \right.} & (4) \\ {{R = {\sum\limits_{i = 1}^{N - 1}R_{i}}}{where}} & (5) \\ {R_{i} = {\sum\limits_{j = {i + 1}}^{N}h_{ij}}} & (6) \end{matrix}$

In the same data set, that Kendall gave, (4, 7, 2, 10, 3, 6, 8, 1, 5, 9), for the first number, 4, there are six numbers on its right which are larger than it. For the second number, 7, there are three, and so on. Thus, the reverse arrangements R_(i) so obtained by Eq. (6) are (6, 3, 6, 0, 4, 2, 1, 2, 1). Therefore, by Eq. (5), the total arrangement R=25 where the minimum & the maximum R is 0 and N(N−1)/2, respectively.

Kendall pointed out the relationship between the actual score Σ and the total reverse arrangement R is

$\begin{matrix} {\Sigma = {{2R} - \frac{N\left( {N - 1} \right)}{2}}} & (7) \end{matrix}$

Kendall also derived the frequency distribution for the actual score Σ under the hypothesis of randomness of these N observations. For simplicity, only the frequency of N up to 6 from Kandall's Table 1 [4] is quoted in Table 1, which has been provided in FIG. 2. As shown, Table 1 illustrates a frequency distribution of Σ for values of N from 1 to 6 (only the positive half of the symmetrical distributions are shown; from Kandall's Table 1 [4])

Under the hypothesis of randomness, R is a random variable. Mann [7] proved its mean and variance as in Eq. (8) & (9), respectively.

$\begin{matrix} {\mu_{R} = \frac{N\left( {N - 1} \right)}{4}} & (8) \\ {o_{R}^{2} = \frac{{2N^{3}} + {3N^{2}} - {5N}}{72}} & (9) \end{matrix}$

Mann listed a table on the probability of obtaining a permutation with R≦ R in permutations of N variables for N=3, . . . , 10. For simplicity, Mann's table with N=3, 4, 5, 6 is duplicated in Table 2, which has provided in FIG. 3.

For N equal or larger than 10, Mann derived Eq. (10) for the accumulated probability.

$\begin{matrix} {{{P(c)} = {\frac{1}{\sqrt{2\pi}}{\int_{- \infty}^{- c}{^{{- x^{2}}/2}{x}}}}}{where}{c = {{{\left( {\mu_{R} - R - \frac{1}{2}} \right)/\sigma_{R}}\mspace{20mu} {for}\mspace{14mu} N} \geq 10.}}} & (10) \end{matrix}$

Mann's paper was referred by Kendall as the first one to recognize that a rank correlation statistic could be used to test randomness as well as independence [6]. Mann discussed the distribution of total reverse arrangement R and proved that the limit distribution of R is normal. The correctness of the tabulated values is important for the application on practical trend tests since many papers and books refereed to this original paper for calculations. The trend test is also widely used in reliability, e.g., for reparable systems, and Mann's RAT along with his tables are frequently referred by reliability statisticians (such as Ansell in his book, Ref. [7]). However, we found some errors in Mann's table (Table 1 of page 246 in Ref. [5]). The errors (the eight numbers for R≧7 when N=6) are shown in italic in Table 2. An obvious error of these is that it is supposed to be 1 when R=Rmax=15, not 0.999 because it shall include all possibilities. The value for R≦8 should not be the same for R≦7. And, the value for R≦9 in Mann's table should be for R≦8, and that for R≦10 should be for R≦9, and so on. For N=7, there are similar mistakes in Mann's original table (i.e., Table 1 of page 246 in Ref. [5]). We re-generate Mann's table for N up to 10 starting from the frequency of Σ given by Kendall's Table 1 and extend the table up to N=12 for both frequency of Σ (extending Kendall's table for n=10, 11, 12) and the accumulated probability for R (extending Mann's Table 1 of page 246). The newly generated tables are shown below for N=3 to 6, respectively, and the tables for N=7 to 12 are in Appendix. The numbers in boldface in Table 3 (for N=6) (see below) are the ones being corrected.

TABLE 3 The accumulated frequency and probability for each R and Σ with N = 3 to 6. The numbers in italic (R = 8 to 15, for N = 6) are what being corrected from Mann's Table 1. Frequency Accumulated R Σ Prob. (t <= T) of Σ Frequency N = 3 0 −3 0.16666667 1 6 1 −1 0.50000000 2 1 2 1 0.83333333 2 3 3 3 1.00000000 1 5 N = 4 0 −6 0.04166667 1 1 1 −4 0.16666667 3 4 2 −2 0.37500000 5 9 3 0 0.62500000 6 15 4 2 0.83333333 5 20 5 4 0.95833333 3 23 6 6 1.00000000 1 24 N = 5 0 −10 0.00833333 1 1 1 −8 0.04166667 4 5 2 −6 0.11666667 9 14 3 −4 0.24166667 15 29 4 −2 0.40833333 20 49 5 0 0.59166667 22 71 6 2 0.75833333 20 91 7 4 0.88333333 15 106 8 6 0.95833333 9 115 9 8 0.99166667 4 119 10 10 1 1 120 N = 6 0 −15 0.00138889 1 1 1 −13 0.00833333 5 6 2 −11 0.02777778 14 20 3 −9 0.06805556 29 49 4 −7 0.13611111 49 98 5 −5 0.23472222 71 169 6 −3 0.35972222 90 259 7 −1 0.50000000 101 360 8 1 0.64027778 101 461 9 3 0.76527778 90 551 10 5 0.86388889 71 622 11 7 0.93194444 49 671 12 9 0.97222222 29 700 13 11 0.99166667 14 714 14 13 0.99861111 5 719 15 15 1.00000000 1 720

Example 1 & 2 depict the trend tests using Table 3 above.

EXAMPLE 1

Consider eight observations: 1, 3, 2, 4, 5, 7, 6, and 8. By Eq. (5) & (6), we have Ri=(7, 5, 5, 4, 3, 1, 1), and total R=26. From the tables in Appendix, the probability when R is equal or higher than 26 is only 1−0.99913194=0.0008681. Therefore, we conclude there is 99.91% confidence that there is an increasing trend.

Example 2 is the reverse order of example 1. That is, the eight observations are (8, 6, 7, 5, 4, 2, 3, 1). We have Ri=(0, 1, 0, 0, 0, 1, 0) and total R=2. The total R can be obtained by another approach, i.e., Rmax−R (when originally ordered as in Example 1). We know Rmax=N(N−1)/2=8 (8−1)/2=28 and, therefore, the total R in Example 2 is =28−26 =2.

From the table in Appendix, the probability when R is equal to or lower than 2 is only 0.0008681 providing the null hypothesis of random variation is valid. Therefore, we have 99.91% confidence that there is a decreasing trend. In other words, the false alarm rate is only 0.08681%, which is much smaller than 5%. We have identified the above. Depending upon the embodiment, we have also provided application of RAT in SPC Run Tests, which have been explained more fully below.

The famous run test of 6 consecutive increasing or decreasing points, introduced by Nelson [3], has been widely used in SPC practices. However, Nelson did not provide the false alarm rate for this test, which is important for SPC practitioners. These 6 points include the base point (i.e., the first point) and, therefore, its false alarm rate is 0.00138889 according to the table for N=6 in Appendix. Some SPC practitioners use 7 consecutive points in their trend test, such as Brook Automation's SPC software FACTORYworks 2.4 [8]. The false alarm rate is even lower to be 0.00019841 from the table N=7 in Appendix. People have slightly different definition on the number of points under trend test. In Smith's SPC book [9], his 7 monotonic increasing or decreasing points do not include the first base point. That is, there are actually 8 points based on Nelson's definition [3] and the false alarm rate is pretty small (=0.0000248, from the table N=8 in Appendix while Smith gave the estimation of the upper limit of 0.008). A non-monotonic trend test for a particular case is introduced in Smith's SPC book [9]: 10 out 11 points are climbing or falling, whose false alarm rate is very small (=0.000902; whereas Smith only provided a rough estimation, 0.0054, in Ref. [9]). As we know, an accurate false alarm rate is very important for resources allocations. A too-high false alarm rate leads to unnecessary investment (on both time and cost) for troubleshooting. On other hand, a too-low false alarm rate results in insufficient sensitivity to detect nonconformance. For non-monotonic trend tests using RAT, we select the false alarm rate equal to or less than that of Nelson's 6 monotonic increasing or decreasing trend test (0.00139); this rate also meet the needs of most applications. Table 4 lists the minimum and maximum total reverse arrangements for up and down trend test and their corresponding p-values (i.e., the false alarm rate), See FIG. 4.

EXAMPLE 3

There are 57 points with central level at 12 and the 3-σ lower & upper control limits (LCL & UCL) at 9 & 15, respectively (See FIG. 5). The total reverse arrangement of the last 7 points is 20 and the false alarm rate is 0.1388. That is, we have 99.86% confidence there is an increasing trend from these 7 points, which cannot be detected by the conventional 6 monotonic increasing trend test since the third point in the circle makes the increasing trend non-monotonic. This control chart actually passes the following three run tests: 1) one point out of 3 sigma, 2) 2 of consecutive 3 points beyond 2 sigma, and 3) 4 of consecutive 5 points beyond 1 sigma. That is, a nonconformance is easily overlooked if we do not apply RAT tests.

EXAMPLE 4

This example is a real case from IC manufacturing (See FIG. 6). A target 300 A SiN (silicon nitride) film is to be deposited by a DCVD (dielectrics chemical vapor deposition) tool. The film thickness is measured by a thin film metrology too and is monitored by an SPC chart. The metrology tool's Xe (Xenon) lamp light intensity happened to degrade gradually and hence affected SiN film thickness.

There are 17 points in FIG. 6 and the last 6 points constitute monotonic increasing trend, which can be detected by the traditional trend test. However, actually, such increasing trend should be detected much earlier from the first 11 points, which show a non-monotonic trend. The total reverse arrangement for the circled 11 points is 48 and the false alarm rate is 0.00038 (from table N=11 in Appendix). Without the RAT tests, the degraded Xe lamp light intensity cannot be detected much earlier as shown in this example. This may lead to serious low-yield events and scrap.

EXAMPLE 5

This example (See FIG. 7) gives the RAT application on WLRC (Wafer Level Reliability Control) in-line monitors. A periodical reliability test of iso-EM (isothermal electromigration) is to monitor metal performance. It is found that the latest 8 measured iso-EM lifetime data (T_(50%)) have non-monotonic increasing trend.

The conventional 6 monotonic increasing trend test rule cannot detect the trend because the third point is slightly lower than the second one (see the circle in FIG. 7). The 7 non-monotonic increasing points inside the circle have total reverse arrangement of R=20 therefore the false alarm rate is 0.00139 (from the table in Appendix). If we include the last point (i.e., the 8^(th) point, which is outside the circle) in the calculation, we have R=27 and the corresponding false alarm rate is 0.000198, which is much lower than that from the 7 points (=0.00139). This is reasonable as we have more evidence of such increasing trend. Although the last 6 points are monotonic increasing and that can be detected by the conventional trend test, it is especially crucial for semiconductor manufacturing to identify nonconformance earlier so as to take early actions to fix problems and reduce possible loss, which is comparably higher than other industries. And, our proposed RAT test again shows its superiority over the traditional trend test on earlier detections of the non-monotonic trends.

Heuristic: In SPC practice, we propose to replace the current monotonic trend test with RAT. A computer code is written to automatically implement this at real time and trigger in-line warning if any. The flow chart in FIG. 8 demonstrates our proposed heuristic. The RAT test applies to the latest 6 points first using the criteria in Table 4. If an increasing or decreasing trend is detected, the program stops. Otherwise, the latest 7 points are tested again with RAT (using Table 4). Such loops continue until N (N=2 in FIG. 8) points are tested by the proposed RAT scheme. Whether we shall continue the RAT test for more data points depends on the frequency of data accumulation and on engineering judgments.

It is necessary to point out a very important flaw with this RAT test at a special case when all points under test are identical although the possibility of having such data is very low. According to the definition of reverse arrangement, the total R will be the same as the monotonic decreasing trend, i.e., R=0, which is the possible minimum total reverse arrangement. The RAT will conclude a decreasing trend with false alarm rate same as the monotonic decreasing trend. Obviously, this is a wrong judgment. However, no paper or book on RAT pointed out such flaw. Fortunately, the probability that all observations are identical is extremely low for continuous normal distribution. A straightforward engineering approach to avoid this special case is to first check the control chart for ties. Moreover, we should always keep sufficient number of significant figures of raw data from measurement, which is determined by the precision of measurement. For effective and automatic RAT tests, we should not round off raw data so we have sufficiently precise data to avoid this wrong judgment. In our computer programs, we have checks on the ties. Moreover, if ties occur frequently, we must check if the measurement and data recording are adequate.

In SPC practice, it is desired to establish whether a sequence of observations is statistically trending up or down. The current widely used monotonic increasing or decreasing trend test cannot detect non-monotonic trends. This paper proposes to use RAT to fulfill the needs of detecting non-monotonic trends. The original papers of RAT were detailed reviewed. We identify mistakes and extend the calculations. We replace the current monotonic trend tests by RAT on semiconductor manufacturing and prevent many potential discrepancies on quality as well as on reliability. Real examples on in-line monitors and reliability applications are reported. A heuristic is illustrated for our proposed RAT test procedure, which is successfully implemented by computer codes for automatic detections.

Depending upon the specific embodiment, the system is overseen and controlled by one or more computer systems, including a microprocessor and/controllers. In a preferred embodiment, the computer system or systems include a common bus, oversees and performs operation and processing of information. The system also has a display 121, which can be a computer display, coupled to the control system 380, which will be described in more detail below. Of course, there can be other modifications, alternatives, and variations. Further details of the present system are provided throughout the specification and more particularly below.

FIG. 9 is a simplified diagram of a computer system 900 that is used to oversee the method of FIG. 1 according to an embodiment of the present invention. This diagram is merely an example, which should not unduly limit the scope of the claims herein. One of ordinary skill in the art would recognize many other modifications, alternatives, and variations. As shown, the computer system includes display device, display screen, cabinet, keyboard, scanner and mouse. Mouse and keyboard are representative “user input devices.” Mouse includes buttons for selection of buttons on a graphical user interface device. Other examples of user input devices are a touch screen, light pen, track ball, data glove, microphone, and so forth.

The system is merely representative of but one type of system for embodying the present invention. It will be readily apparent to one of ordinary skill in the art that many system types and configurations are suitable for use in conjunction with the present invention. In a preferred embodiment, computer system 900 includes a Pentium™ class based computer, running Windows™ NT operating system by Microsoft Corporation or Linux based systems from a variety of sources. However, the system is easily adapted to other operating systems and architectures by those of ordinary skill in the art without departing from the scope of the present invention. As noted, mouse can have one or more buttons such as buttons. Cabinet houses familiar computer components such as disk drives, a processor, storage device, etc. Storage devices include, but are not limited to, disk drives, magnetic tape, solid-state memory, flash memory, bubble memory, etc. Cabinet can include additional hardware such as input/output (I/O) interface cards for connecting computer system to external devices external storage, other computers or additional peripherals, which are further described below.

FIG. 10 is a more detailed diagram of hardware elements in the computer system according to an embodiment of the present invention. This diagram is merely an example, which should not unduly limit the scope of the claims herein. One of ordinary skill in the art would recognize many other modifications, alternatives, and variations. As shown, basic subsystems are included in computer system 900. In specific embodiments, the subsystems are interconnected via a system bus 1385. Additional subsystems such as a printer 1384, keyboard 1388, fixed disk 1389, monitor 1386, which is coupled to display adapter 1392, and others are shown. Peripherals and input/output (I/O) devices, which couple to I/O controller 1381, can be connected to the computer system by any number of means known in the art, such as serial port 1387. For example, serial port 1387 can be used to connect the computer system to a modem 1391, which in turn connects to a wide area network such as the Internet, a mouse input device, or a scanner. The interconnection via system bus allows central processor 1383 to communicate with each subsystem and to control the execution of instructions from system memory 1382 or the fixed disk 1389, as well as the exchange of information between subsystems. Other arrangements of subsystems and interconnections are readily achievable by those of ordinary skill in the art. System memory, and the fixed disk are examples of tangible media for storage of computer programs, other types of tangible media include floppy disks, removable hard disks, optical storage media such as CD-ROMS and bar codes, and semiconductor memories such as flash memory, read-only-memories (ROM), and battery backed memory.

Although the above has been illustrated in terms of specific hardware features, it would be recognized that many variations, alternatives, and modifications can exist. For example, any of the hardware features can be further combined, or even separated. The features can also be implemented, in part, through software or a combination of hardware and software. The hardware and software can be further integrated or less integrated depending upon the application. Further details of certain methods according to the present invention can be found throughout the present specification and more particularly below.

REFERENCES

-   1. Shewhart, W. A. (edited and new foreword by Deming, W. E.),     Statistical Methods from the Viewpoint of Quality Control, Dover     Publications, 1986, New York, USA. -   2. Western Electric, Statistical Quality Control Handbook, 1958. -   3. Nelson, Lloyd S., “The Shewhart Control Chart-Test of Special     Causes,” J of Quality Technology, 16(4), 1984, pp. 237-239. -   4. M. G. Kendall, “A New Measure of rank Correlation”, Biometrika,     Vol. 30, pp. 81-93, June 1938. -   5. Henry B. Mann, “Nonparametric Tests Against Trend”, Econometrica,     Vol. 13, No. 3, July, 1945, pp. 245-259. -   6. M. G. Kendall, The advanced Theory of Statistics, Vol. 2,     Inference and Relationship, 1961. -   7. J. I. Ansell and M. J. Phillips, “Practical Methods for     Reliability Data Analysis”, Oxford Statistical Science Series,     Oxford Clarendon Press, p. 142, 1994. -   8. FACTORYworks 2.4, BROOKS Automation Inc, 15 Elizabeth Drive,     Chelmsford, Mass. 01824, USA., www.brooks.com. -   9. Gerald M. Smith, “Statistical Process Control and Quality     Improvement”, 4^(th) Edition, Prentice Hall, p. 394-398, 2001.

APPENDIX Accumulated frequency and probability for R and Σ for N = 7 to 12 Frequency R Σ Prob. (t <= T) of Σ Accumulated Frequency N = 7 0 −21 0.00019841 1 1 1 −19 0.00138889 6 7 2 −17 0.00535714 20 27 3 −15 0.01507937 49 76 4 −13 0.03452381 98 174 5 −11 0.06805556 169 343 6 −9 0.11944444 259 602 7 −7 0.19067460 359 961 8 −5 0.28095238 455 1416 9 −3 0.38630952 531 1947 10 −1 0.50000000 573 2520 11 1 0.61369048 573 3093 12 3 0.71904762 531 3624 13 5 0.80932540 455 4079 14 7 0.88055556 359 4438 15 9 0.93194444 259 4697 16 11 0.96547619 169 4866 17 13 0.98492063 98 4964 18 15 0.99464286 49 5013 19 17 0.99861111 20 5033 20 19 0.99980159 6 5039 21 21 1.00000000 1 5040 N = 8 0 −28 0.00002480 1 1 1 −26 0.00019841 7 8 2 −24 0.00086806 27 35 3 −22 0.00275298 76 111 4 −20 0.00706845 174 285 5 −18 0.01557540 343 628 6 −16 0.03050595 602 1230 7 −14 0.05434028 961 2191 8 −12 0.08943452 1415 3606 9 −10 0.13754960 1940 5546 10 −8 0.19937996 2493 8039 11 −6 0.27420635 3017 11056 12 −4 0.35977183 3450 14506 13 −2 0.45243056 3736 18242 14 0 0.54756944 3836 22078 15 2 0.64022817 3736 25814 16 4 0.72579365 3450 29264 17 6 0.80062004 3017 32281 18 8 0.86245040 2493 34774 19 10 0.91056548 1940 36714 20 12 0.94565972 1415 38129 21 14 0.96949405 961 39090 22 16 0.98442460 602 39692 23 18 0.99293155 343 40035 24 20 0.99724702 174 40209 25 22 0.99913194 76 40285 26 24 0.99980159 27 40312 27 26 0.99997520 7 40319 28 28 1.00000000 1 40320 N = 9 0 −36 0.00000276 1 1 1 −34 0.00002480 8 9 2 −32 0.00012125 35 44 3 −30 0.00042714 111 155 4 −28 0.00121252 285 440 5 −26 0.00294312 628 1068 6 −24 0.00633267 1230 2298 7 −22 0.01237048 2191 4489 8 −20 0.02230765 3606 8095 9 −18 0.03758818 5545 13640 10 −16 0.05971947 8031 21671 11 −14 0.09009039 11021 32692 12 −12 0.12975915 14395 47087 13 −10 0.17924383 17957 65044 14 −8 0.23835428 21450 86494 15 −6 0.30610119 24584 111078 16 −4 0.38070712 27073 138151 17 −2 0.45972773 28675 166826 18 0 0.54027227 29228 196054 19 2 0.61929288 28675 224729 20 4 0.69389881 27073 251802 21 6 0.76164572 24584 276386 22 8 0.82075617 21450 297836 23 10 0.87024085 17957 315793 24 12 0.90990961 14395 330188 25 14 0.94028053 11021 341209 26 16 0.96241182 8031 349240 27 18 0.97769235 5545 354785 28 20 0.98762952 3606 358391 29 22 0.99366733 2191 360582 30 24 0.99705688 1230 361812 31 26 0.99878748 628 362440 32 28 0.99957286 285 362725 33 30 0.99987875 111 362836 34 32 0.99997520 35 362871 35 34 0.99999724 8 362879 36 36 1.00000000 1 362880 N = 10 0 −45 0.00000028 1 1 1 −43 0.00000276 9 10 2 −41 0.00001488 44 54 3 −39 0.00005759 155 209 4 −37 0.00017885 440 649 5 −35 0.00047316 1068 1717 6 −33 0.00110643 2298 4015 7 −31 0.00234347 4489 8504 8 −29 0.00457424 8095 16599 9 −27 0.00833306 13640 30239 10 −25 0.01430473 21670 51909 11 −23 0.02331129 32683 84592 12 −21 0.03627508 47043 131635 13 −19 0.05415675 64889 196524 14 −17 0.07787092 86054 282578 15 −15 0.10818673 110010 392588 16 −13 0.14562417 135853 528441 17 −11 0.19035990 162337 690778 18 −9 0.24215636 187959 878737 19 −7 0.30032683 211089 1089826 20 −5 0.36374476 230131 1319957 21 −3 0.43090030 243694 1563651 22 −1 0.50000000 250749 1814400 23 1 0.56909970 250749 2065149 24 3 0.63625524 243694 2308843 25 5 0.69967317 230131 2538974 26 7 0.75784364 211089 2750063 27 9 0.80964010 187959 2938022 28 11 0.85437583 162337 3100359 29 13 0.89181327 135853 3236212 30 15 0.92212908 110010 3346222 31 17 0.94584325 86054 3432276 32 19 0.96372492 64889 3497165 33 21 0.97668871 47043 3544208 34 23 0.98569527 32683 3576891 35 25 0.99166694 21670 3598561 36 27 0.99542576 13640 3612201 37 29 0.99765653 8095 3620296 38 31 0.99889357 4489 3624785 39 33 0.99952684 2298 3627083 40 35 0.99982115 1068 3628151 41 37 0.99994241 440 3628591 42 39 0.99998512 155 3628746 43 41 0.99999724 44 3628790 44 43 0.99999972 9 3628799 45 45 1.00000000 1 3628800 N = 11 0 −55 0.00000003 1 1 1 −53 0.00000028 10 11 2 −51 0.00000163 54 65 3 −49 0.00000686 209 274 4 −47 0.00002312 649 923 5 −45 0.00006614 1717 2640 6 −43 0.00016672 4015 6655 7 −41 0.00037976 8504 15159 8 −39 0.00079560 16599 31758 9 −37 0.00155316 30239 61997 10 −35 0.00285359 51909 113906 11 −33 0.00497277 84591 198497 12 −31 0.00827025 131625 330122 13 −29 0.01319224 196470 526592 14 −27 0.02026618 282369 808961 15 −25 0.03008508 391939 1200900 16 −23 0.04328062 526724 1727624 17 −21 0.06048548 686763 2414387 18 −19 0.08228666 870233 3284620 19 −17 0.10917326 1073227 4357847 20 −15 0.14148341 1289718 5647565 21 −13 0.17935573 1511742 7159307 22 −11 0.22269107 1729808 8889115 23 −9 0.27112967 1933514 10822629 24 −7 0.32404772 2112319 12934948 25 −5 0.38057520 2256396 15191344 26 −3 0.43963492 2357475 17548819 27 −1 0.50000000 2409581 19958400 28 1 0.56036508 2409581 22367981 29 3 0.61942480 2357475 24725456 30 5 0.67595228 2256396 26981852 31 7 0.72887033 2112319 29094171 32 9 0.77730893 1933514 31027685 33 11 0.82064427 1729808 32757493 34 13 0.85851659 1511742 34269235 35 15 0.89082674 1289718 35558953 36 17 0.91771334 1073227 36632180 37 19 0.93951452 870233 37502413 38 21 0.95671938 686763 38189176 39 23 0.96991492 526724 38715900 40 25 0.97973382 391939 39107839 41 27 0.98680776 282369 39390208 42 29 0.99172975 196470 39586678 43 31 0.99502723 131625 39718303 44 33 0.99714641 84591 39802894 45 35 0.99844684 51909 39854803 46 37 0.99920440 30239 39885042 47 39 0.99962024 16599 39901641 48 41 0.99983328 8504 39910145 49 43 0.99993386 4015 39914160 50 45 0.99997688 1717 39915877 51 47 0.99999314 649 39916526 52 49 0.99999837 209 39916735 53 51 0.99999972 54 39916789 54 53 0.99999997 10 39916799 55 55 1.00000000 1 39916800 N = 12 0 −66 0.00000000 1 1 1 −64 0.00000003 11 12 2 −62 0.00000016 65 77 3 −60 0.00000073 274 351 4 −58 0.00000266 923 1274 5 −56 0.00000817 2640 3914 6 −54 0.00002206 6655 10569 7 −52 0.00005371 15159 25728 8 −50 0.00012001 31758 57486 9 −48 0.00024944 61997 119483 10 −46 0.00048724 113906 233389 11 −44 0.00090164 198497 431886 12 −42 0.00159082 330121 762007 13 −40 0.00269015 526581 1288588 14 −38 0.00437887 808896 2097484 15 −36 0.00688538 1200626 3298110 16 −34 0.01049018 1726701 5024811 17 −32 0.01552512 2411747 7436558 18 −30 0.02236845 3277965 10714523 19 −28 0.03143457 4342688 15057211 20 −26 0.04315856 5615807 20673018 21 −24 0.05797544 7097310 27770328 22 −22 0.07629523 8775209 36545537 23 −20 0.09847497 10624132 47169669 24 −18 0.12478976 12604826 59774495 25 −16 0.15540501 14664752 74439247 26 −14 0.19035240 16739858 91179105 27 −12 0.22951198 18757500 109936605 28 −10 0.27260235 20640357 130576962 29 −8 0.31918063 22311069 152888031 30 −6 0.36865276 23697232 176585263 31 −4 0.42029418 24736324 201321587 32 −2 0.47327964 25380120 226701707 33 0 0.52672036 25598186 252299893 34 2 0.57970582 25380120 277680013 35 4 0.63134724 24736324 302416337 36 6 0.68081937 23697232 326113569 37 8 0.72739765 22311069 348424638 38 10 0.77048802 20640357 369064995 39 12 0.80964760 18757500 387822495 40 14 0.84459499 16739858 404562353 41 16 0.87521024 14664752 419227105 42 18 0.90152503 12604826 431831931 43 20 0.92370477 10624132 442456063 44 22 0.94202456 8775209 451231272 45 24 0.95684144 7097310 458328582 46 26 0.96856543 5615807 463944389 47 28 0.97763155 4342688 468287077 48 30 0.98447488 3277965 471565042 49 32 0.98950982 2411747 473976789 50 34 0.99311462 1726701 475703490 51 36 0.99562113 1200626 476904116 52 38 0.99730985 808896 477713012 53 40 0.99840918 526581 478239593 54 42 0.99909836 330121 478569714 55 44 0.99951276 198497 478768211 56 46 0.99975056 113906 478882117 57 48 0.99987999 61997 478944114 58 50 0.99994629 31758 478975872 59 52 0.99997794 15159 478991031 60 54 0.99999183 6655 478997686 61 56 0.99999734 2640 479000326 62 58 0.99999927 923 479001249 63 60 0.99999984 274 479001523 64 62 0.99999997 65 479001588 65 64 1.00000000 11 479001599 66 66 1.00000000 1 479001600

The disclosures and the description herein are purely illustrative and are not to be limited with the above examples. A person skilled in reliability engineering and reliability statistics would be able to apply the method disclosed in the above embodiments to his/her particular product, component or system in reliability testing. It is also understood that the examples and embodiments described herein are for illustrative purposes only and that various modifications or changes in light thereof will be suggested to persons skilled in the art and are to be included within the spirit and purview of this application and scope of the appended claims. 

1. A method for manufacturing semiconductor devices, the method comprising: providing a process associated with a manufacture of a semiconductor device; collecting a plurality information having a non-monotonic trend of at least one parameter associated with the process over a determined period; processing the plurality of information having the non-monotonic trend; detecting an increasing or a decreasing trend from the processed plurality of information having the non-monotonic trend; and performing an action based upon at least the detected increasing or decreasing trend.
 2. The method of claim 1 wherein the determined period comprises a time period.
 3. The method of claim 1 wherein the determined period comprises a spatial frequency.
 4. The method of claim 1 wherein the processing and detecting comprises a reverse arrangement test.
 5. The method of claim 1 wherein the processing and detecting comprises a randomness test.
 6. The method of claim 1 wherein the processing and detecting is called a Cox-Stuart test.
 7. The method of claim 1 wherein the reverse arrangement test comprises determining a number of data points, process the plurality of information for the data points.
 8. The method of claim 7 further comprising adding the number of data points by one, and processing the plurality of data points.
 9. The method of claim 8 further comprising reporting no change.
 10. The method of claim 1 further comprising associating the increasing or decreasing trend with a predetermined false alarm rate.
 11. The method of claim 10 wherein if the trend is within the predetermined false alarm rate, output a decreasing or increasing rate.
 12. The method of claim 10 wherein if the trend is outside of the predetermined false alarm rate, add one more sample and continue to process.
 13. A system for manufacturing semiconductor devices, the system comprising one or more memories, the one or more memories: one or more codes directed to initiating a process associated with a manufacture of a semiconductor device; one or more codes directed to collecting a plurality information having a non-monotonic trend of at least one parameter associated with the process over a determined period; one or more codes directed to processing the plurality of information having the non-monotonic trend; one or more codes directed to detecting an increasing or a decreasing trend from the processed plurality of information having the non-monotonic trend; and one or more codes directed to outputting a code to perform an action based upon at least the detected increasing or decreasing trend.
 14. The system of claim 13 wherein the determined period comprises a time period.
 15. The system of claim 13 wherein the determined period comprises a spatial frequency.
 16. The system of claim 13 wherein the processing and detecting comprises a reverse arrangement test.
 17. The system of claim 13 wherein the processing and detecting comprises a randomness test.
 18. The system of claim 13 wherein the processing and detecting is called a Cox-Stuart test.
 19. The system of claim 13 wherein the reverse arrangement test comprises determining a number of data points, process the plurality of information for the data points.
 20. The system of claim 19 further comprising adding the number of data points by one, and processing the plurality of data points.
 21. The system of claim 20 further comprising reporting no change.
 22. The system of claim 21 further comprising associating the increasing or decreasing trend with a predetermined false alarm rate.
 23. The system of claim 22 wherein if the trend is within the predetermined false alarm rate, output a decreasing or increasing rate.
 24. The system of claim 23 wherein if the trend is outside of the predetermined false alarm rate, add one more sample and continue to process. 